Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128836 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1762 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #440 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,11,7,12,6,10)(3,13,14,16,15,4,9,8)(17,18,21,26)(19,29)(20,34,36,27)(22,39,37,25)(23,40,38,28)(24,35)(30,45)(31,43)(32,41)(33,44)(42,46)(47,48);(1,3,6,9,7,15,5,14)(2,8,10,4,12,16,11,13)(17,19,22,32,38,46,36,31,21,33,37,47,23,30,20,35)(18,24,27,42,40,48,39,29,26,43,34,45,28,41,25,44);(1,4,5,8,7,13,6,16)(2,9,11,3,12,14,10,15)(19,30,33,46)(20,36)(22,37)(24,41,43,48)(25,39)(27,34)(29,45,44,42)(31,47,35,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=g,e^2=hi,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=cgh,d^b=dh,e^b=ei,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285666, #72285993, #72287417, … |
| maximal normal subgroup of | #72285666, #72285993, #72287417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36230);